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AQA Further Paper 3 Statistics 2022 June Q2
1 marks Moderate -0.8
2 The random variable \(X\) has probability density function $$f ( x ) = \begin{cases} 1 & 0 < x \leq \frac { 1 } { 2 } \\ \frac { 3 } { 8 } x ^ { - 2 } & \frac { 1 } { 2 } < x \leq \frac { 3 } { 2 } \\ 0 & \text { otherwise } \end{cases}$$ Find \(\mathrm { P } ( X < 1 )\) Circle your answer.
[0pt] [1 mark] \(\frac { 1 } { 8 }\) \(\frac { 3 } { 8 }\) \(\frac { 5 } { 8 }\) \(\frac { 7 } { 8 }\) \includegraphics[max width=\textwidth, alt={}, center]{62cee897-6eac-40b3-84c1-a0d165ba6903-03_2488_1718_219_153}
AQA Further Paper 3 Statistics 2022 June Q3
5 marks Moderate -0.8
3 The random variable \(X\) has an exponential distribution with probability density function \(\mathrm { f } ( x ) = \lambda \mathrm { e } ^ { - \lambda x }\) where \(x \geq 0\) 3
  1. Show that the cumulative distribution function, for \(x \geq 0\), is given by \(\mathrm { F } ( x ) = 1 - \mathrm { e } ^ { - \lambda x }\) [0pt] [3 marks]
    3
  2. Given that \(\lambda = 2\), find \(\mathrm { P } ( X > 1 )\), giving your answer to three decimal places.
AQA Further Paper 3 Statistics 2022 June Q4
5 marks Standard +0.3
4 Daisies and dandelions are the only flowers growing in a field. The number of daisies per square metre in the field has a mean of 16
The number of dandelions per square metre in the field has a mean of 10
The number of daisies per square metre and the number of dandelions per square metre are independent. 4
  1. Using a Poisson model, find the probability that a randomly selected square metre from the field has a total of at least 30 flowers, giving your answer to three decimal places.
    4
  2. A survey of the entire field is taken.
    The standard deviation of the total number of flowers per square metre is 10 State, with a reason, whether the model used in part (a) is valid.
AQA Further Paper 3 Statistics 2022 June Q5
6 marks Standard +0.3
5 The mass, \(X\), in grams of a particular type of apple is modelled using a normal distribution. A random sample of 12 apples is collected and the summarised results are $$\sum x = 1038 \quad \text { and } \quad \sum x ^ { 2 } = 90100$$ 5
  1. A 99\% confidence interval for the population mean of the masses of the apples is constructed using the random sample. Show that the confidence interval is \(( 81.7,91.3 )\) with values correct to three significant figures.
    5
  2. Padraig claims that the population mean mass of the apples is 85 grams. He carries out a hypothesis test at the \(1 \%\) level of significance using the random sample of 12 apples. The hypotheses are $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 85 \\ & \mathrm { H } _ { 1 } : \mu \neq 85 \end{aligned}$$ State, with a reason, whether the null hypothesis is accepted or rejected.
    5
  3. Interpret, in context, the conclusion to the hypothesis test in part (b).
AQA Further Paper 3 Statistics 2022 June Q6
8 marks Standard +0.3
6 The discrete random variable \(X\) has probability distribution function $$\mathrm { P } ( X = x ) = \begin{cases} a & x = 0 \\ b & x = 1 \\ c & x = 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(a , b\) and \(c\) are constants.
The mean of \(X\) is 1.2 and the variance of \(X\) is 0.56
6
  1. Deduce the values of \(a , b\) and \(c\) 6
  2. The continuous random variable \(Y\) is independent of \(X\) and has variance 15 Find \(\operatorname { Var } ( X - 2 Y - 11 )\) [0pt] [2 marks]
AQA Further Paper 3 Statistics 2022 June Q7
9 marks Easy -1.2
7
  1. Test the scientist's claim, using the 10\% level of significance.
    7
  2. For the context of the test carried out in part (a), state the meaning of a Type I error. [1 mark]
AQA Further Paper 3 Statistics 2022 June Q8
11 marks Standard +0.3
8 The continuous random variable \(X\) has cumulative distribution function \(\mathrm { F } ( x )\) where $$\mathrm { F } ( x ) = \begin{cases} 0 & x = 0 \\ \mathrm { e } ^ { k x } - 1 & 0 \leq x \leq 5 \\ 1 & x > 5 \end{cases}$$ 8
  1. Show that \(k = \frac { 1 } { 5 } \ln 2\) [0pt] [2 marks]
    8
  2. Show that the median of \(X\) is \(a \frac { \ln b } { \ln 2 } - c\), where \(a , b\) and \(c\) are integers to be found.
    8
  		3. Show that the mean of \(X\) is \(p - \frac { q } { \ln 2 }\), where \(p\) and \(q\) are integers to be found.
AQA Further Paper 3 Statistics 2022 June Q9
4 marks Moderate -0.8
9 Lianne models the maximum time in hours that a rechargeable battery can be used, before needing to be recharged, with a rectangular distribution with values between 8 and 12 9
  1. The probability that the maximum time the battery can be used before needing to be recharged is more than 10.5 hours is equal to \(p\) Lianne will only buy the battery if \(p\) is more than 0.4
    Determine whether Lianne will buy the battery.
    [0pt] [2 marks]
    9
  2. A histogram is plotted for 100 recharges showing the maximum time the battery can be used before needing to be recharged. \includegraphics[max width=\textwidth, alt={}, center]{62cee897-6eac-40b3-84c1-a0d165ba6903-15_670_1186_404_427} Explain why the model used in part (a) may not be valid and suggest the name of a different distribution that could be used to model the maximum time between recharges. \includegraphics[max width=\textwidth, alt={}, center]{62cee897-6eac-40b3-84c1-a0d165ba6903-16_2488_1732_219_139}
    \includegraphics[max width=\textwidth, alt={}]{62cee897-6eac-40b3-84c1-a0d165ba6903-20_2496_1721_214_148}
AQA Further Paper 3 Statistics 2023 June Q1
1 marks Easy -1.8
1 The discrete random variable \(A\) takes only the values 0,2 and 4, and has cumulative distribution function \(\mathrm { F } ( a ) = \mathrm { P } ( A \leq a )\)
\(a\)024
\(\mathrm {~F} ( a )\)0.20.61
Find \(\mathrm { P } ( A = 2 )\) Circle your answer. \(0 \quad 0.4 \quad 0.6 \quad 0.8\)
AQA Further Paper 3 Statistics 2023 June Q2
1 marks Moderate -0.5
2 The time, \(T\) days, between rain showers in a city in autumn can be modelled by an exponential distribution with mean 1.25 Find the distribution of the number of rain showers per day in the city.
Tick ( \(\checkmark\) ) one box.
[0pt] [1 mark] \includegraphics[max width=\textwidth, alt={}, center]{1e2fdd33-afa4-486f-a9e2-1d425ed14eee-03_108_113_1800_370}
DistributionMean
Exponential0.8
\includegraphics[max width=\textwidth, alt={}]{1e2fdd33-afa4-486f-a9e2-1d425ed14eee-03_108_113_1932_370}
Exponential1.25
□
Poisson0.8
□
Poisson1.25
AQA Further Paper 3 Statistics 2023 June Q3
3 marks Moderate -0.8
3 The masses of tins of a particular brand of spaghetti are normally distributed with mean \(\mu\) grams and standard deviation 4.1 grams. A random sample of 11 tins of spaghetti has a mean mass of 401.8 grams.
Construct a \(98 \%\) confidence interval for \(\mu\), giving your values to one decimal place.
AQA Further Paper 3 Statistics 2023 June Q4
5 marks Standard +0.3
4 The random variable \(X\) has a normal distribution with unknown mean \(\mu\) and unknown variance \(\sigma ^ { 2 }\) A random sample of 8 observations of \(X\) has mean \(\bar { x } = 101.5\) and gives the unbiased estimate of the variance as \(s ^ { 2 } = 4.8\) The random sample is used to conduct a hypothesis test at the \(10 \%\) level of significance with the hypotheses $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 100 \\ & \mathrm { H } _ { 1 } : \mu \neq 100 \end{aligned}$$ Carry out the hypothesis test.
AQA Further Paper 3 Statistics 2023 June Q5
8 marks Standard +0.3
5 A school management team oversees 11 different schools.
The school management team allows each student in the schools to choose one enrichment activity from 11 possible activities. The school management team count the number of students in each school choosing each of the possible activities. They then conduct a \(\chi ^ { 2 }\)-test for association with the data they have gathered. 5
  1. Exactly one of the calculated expected frequencies for the \(\chi ^ { 2 }\)-test is less than 5
    Explain why the number of degrees of freedom for the test is 90
    5
  2. The school management team claims that there is an association between the school a student attends and the activity they choose. The test statistic is 124.8 Test the claim using the \(1 \%\) level of significance.
    5
  3. During the hypothesis test, the value of \(\frac { ( O - E ) ^ { 2 } } { E }\), where \(O\) is the observed frequency and \(E\) is the expected frequency, was calculated for each group of students. The values for four groups of students are shown in the table below.
    Group\(\frac { ( O - E ) ^ { 2 } } { E }\)
    Attends school 3 and chose activity 10.01
    Attends school 8 and chose activity 318.5
    Attends school 8 and chose activity 724.2
    Attends school 11 and chose activity 749.0
    State, with a reason, which of the four groups of students represents the strongest source of association.
AQA Further Paper 3 Statistics 2023 June Q6
7 marks Easy -1.2
6 A game consists of two rounds. The first round of the game uses a random number generator to output the score \(X\), a real number between 0 and 10 6
  1. Find \(\mathrm { P } ( X > 4 )\) 6
  2. The second round of the game uses an unbiased dice, with faces numbered 1 to 6 , to give the score \(Y\) The variables \(X\) and \(Y\) are independent.
    6 (b) (i) Find the mean total score of the game.
    6 (b) (ii) Find the variance of the total score of the game.
AQA Further Paper 3 Statistics 2023 June Q7
11 marks Standard +0.3
7 Company \(A\) uses a machine to produce toys. The number of toys in a week that do not pass Company \(A\) 's quality checks is modelled by a Poisson distribution \(X\) with standard deviation 5 The machine producing the toys breaks down.
After it is repaired, 16 toys in the next week do not pass the quality checks.
7
  1. Investigate whether the average number of toys that do not pass the quality checks in a week has changed, using the \(5 \%\) level of significance.
    7
  2. For the test carried out in part (a), state in context the meaning of a Type II error. 7
  3. Company \(B\) uses a different machine to produce toys.
    The number of toys in a week that do not pass Company B's quality checks is modelled by a Poisson distribution \(Y\) with mean 18 The variables \(X\) and \(Y\) are independent.
    Find the distribution of the total number of toys in a week produced by companies \(A\) and \(B\) that do not pass their quality checks. 7
  4. State two reasons why a Poisson distribution may not be a valid model for the number of toys that do not pass the quality checks in a week. Reason 1 \(\_\_\_\_\) Reason 2 \(\_\_\_\_\)
AQA Further Paper 3 Statistics 2023 June Q8
14 marks Standard +0.3
8 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} k \sin 2 x & 0 \leq x \leq \frac { \pi } { 6 } \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant. 8
  1. Show that \(k = 4\) 8
  2. Find the cumulative distribution function \(\mathrm { F } ( x )\) 8
  3. Find the median of \(X\), giving your answer to three significant figures. 8
  4. Find the mean of \(X\) giving your answer in the form \(\frac { 1 } { a } ( b \sqrt { 3 } - \pi )\) where \(a\) and \(b\) are integers. \includegraphics[max width=\textwidth, alt={}, center]{1e2fdd33-afa4-486f-a9e2-1d425ed14eee-14_2492_1721_217_150}
AQA Further Paper 3 Statistics 2024 June Q1
1 marks Easy -1.8
1 The random variable \(X\) has a Poisson distribution with mean 16 Find the standard deviation of \(X\) Circle your answer.
4
8
16
256
AQA Further Paper 3 Statistics 2024 June Q4
6 marks Moderate -0.3
4
8
16
256 2 The random variable \(T\) has an exponential distribution with mean 2 Find \(\mathrm { P } ( T \leq 1.4 )\) Circle your answer. \(\mathrm { e } ^ { - 2.8 }\) \(\mathrm { e } ^ { - 0.7 }\) \(1 - e ^ { - 0.7 }\) \(1 - \mathrm { e } ^ { - 2.8 }\) The continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 2 \\ - \frac { 1 } { 9 } y ^ { 2 } + \frac { 10 } { 9 } y - \frac { 16 } { 9 } & 2 \leq y < 5 \\ 1 & y \geq 5 \end{array} \right.$$ Find the median of \(Y\) Circle your answer. 2 \(\frac { 10 - 3 \sqrt { 2 } } { 2 }\) \(\frac { 7 } { 2 }\) \(\frac { 10 + 3 \sqrt { 2 } } { 2 }\) Turn over for the next question 4 Research has shown that the mean number of volcanic eruptions on Earth each day is 20 Sandra records 162 volcanic eruptions during a period of one week. Sandra claims that there has been an increase in the mean number of volcanic eruptions per week. Test Sandra's claim at the \(5 \%\) level of significance.
AQA Further Paper 3 Statistics 2024 June Q5
5 marks Standard +0.3
5 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 6 } e ^ { \frac { x } { 3 } } & 0 \leq x \leq \ln 27 \\ 0 & \text { otherwise } \end{cases}$$ Show that the mean of \(X\) is \(\frac { 3 } { 2 } ( \ln 27 - 2 )\)
AQA Further Paper 3 Statistics 2024 June Q6
9 marks Standard +0.3
6 Over time it has been accepted that the mean retirement age for professional baseball players is 29.5 years old. Imran claims that the mean retirement age is no longer 29.5 years old.
He takes a random sample of 5 recently retired professional baseball players and records their retirement ages, \(x\). The results are $$\sum x = 152.1 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 7.81$$ 6
  1. State an assumption that you should make about the distribution of the retirement ages to investigate Imran's claim. 6
  2. Investigate Imran's claim, using the 10\% level of significance.
AQA Further Paper 3 Statistics 2024 June Q8
5 marks Moderate -0.3
8
16
256 2 The random variable \(T\) has an exponential distribution with mean 2 Find \(\mathrm { P } ( T \leq 1.4 )\) Circle your answer. \(\mathrm { e } ^ { - 2.8 }\) \(\mathrm { e } ^ { - 0.7 }\) \(1 - e ^ { - 0.7 }\) \(1 - \mathrm { e } ^ { - 2.8 }\) The continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 2 \\ - \frac { 1 } { 9 } y ^ { 2 } + \frac { 10 } { 9 } y - \frac { 16 } { 9 } & 2 \leq y < 5 \\ 1 & y \geq 5 \end{array} \right.$$ Find the median of \(Y\) Circle your answer. 2 \(\frac { 10 - 3 \sqrt { 2 } } { 2 }\) \(\frac { 7 } { 2 }\) \(\frac { 10 + 3 \sqrt { 2 } } { 2 }\) Turn over for the next question 4 Research has shown that the mean number of volcanic eruptions on Earth each day is 20 Sandra records 162 volcanic eruptions during a period of one week. Sandra claims that there has been an increase in the mean number of volcanic eruptions per week. Test Sandra's claim at the \(5 \%\) level of significance.
5 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 6 } e ^ { \frac { x } { 3 } } & 0 \leq x \leq \ln 27 \\ 0 & \text { otherwise } \end{cases}$$ Show that the mean of \(X\) is \(\frac { 3 } { 2 } ( \ln 27 - 2 )\) 6 Over time it has been accepted that the mean retirement age for professional baseball players is 29.5 years old. Imran claims that the mean retirement age is no longer 29.5 years old.
He takes a random sample of 5 recently retired professional baseball players and records their retirement ages, \(x\). The results are $$\sum x = 152.1 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 7.81$$ 6
  1. State an assumption that you should make about the distribution of the retirement ages to investigate Imran's claim. 6
  2. Investigate Imran's claim, using the 10\% level of significance.
AQA Further Paper 3 Statistics 2024 June Q9
11 marks Standard +0.3
9 A company owns three shops, A, B and C, which are based in different towns. Each shop gives a questionnaire to 250 of their customers, and every customer completes the questionnaire. One of the questions asks whether the customer rates the shop as good, satisfactory or poor. For shop A, 26\% of customers rate the shop as good and 38\% of customers rate the shop as poor. For shop B, 32\% of customers rate the shop as good and 40\% of customers rate the shop as satisfactory. Altogether, there are 210 good ratings and 261 satisfactory ratings. 9
  1. Complete the following table with the observed frequencies.
    \multirow{2}{*}{}Rating
    GoodSatisfactoryPoor
    \multirow{3}{*}{Shop}A
    B
    C
    9
  2. Carry out a test for association between shop and rating, using the 1\% level of significance.
AQA Further Paper 3 Statistics 2024 June Q16
Moderate -0.8
16
256 2 The random variable \(T\) has an exponential distribution with mean 2 Find \(\mathrm { P } ( T \leq 1.4 )\) Circle your answer. \(\mathrm { e } ^ { - 2.8 }\) \(\mathrm { e } ^ { - 0.7 }\) \(1 - e ^ { - 0.7 }\) \(1 - \mathrm { e } ^ { - 2.8 }\) The continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 2 \\ - \frac { 1 } { 9 } y ^ { 2 } + \frac { 10 } { 9 } y - \frac { 16 } { 9 } & 2 \leq y < 5 \\ 1 & y \geq 5 \end{array} \right.$$ Find the median of \(Y\) Circle your answer. 2 \(\frac { 10 - 3 \sqrt { 2 } } { 2 }\) \(\frac { 7 } { 2 }\) \(\frac { 10 + 3 \sqrt { 2 } } { 2 }\) Turn over for the next question 4 Research has shown that the mean number of volcanic eruptions on Earth each day is 20 Sandra records 162 volcanic eruptions during a period of one week. Sandra claims that there has been an increase in the mean number of volcanic eruptions per week. Test Sandra's claim at the \(5 \%\) level of significance.
5 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 6 } e ^ { \frac { x } { 3 } } & 0 \leq x \leq \ln 27 \\ 0 & \text { otherwise } \end{cases}$$ Show that the mean of \(X\) is \(\frac { 3 } { 2 } ( \ln 27 - 2 )\) 6 Over time it has been accepted that the mean retirement age for professional baseball players is 29.5 years old. Imran claims that the mean retirement age is no longer 29.5 years old.
He takes a random sample of 5 recently retired professional baseball players and records their retirement ages, \(x\). The results are $$\sum x = 152.1 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 7.81$$ 6
  1. State an assumption that you should make about the distribution of the retirement ages to investigate Imran's claim. 6
  2. Investigate Imran's claim, using the 10\% level of significance.
AQA Further Paper 3 Mechanics 2019 June Q1
1 marks Moderate -0.8
1 A spring has natural length 0.4 metres and modulus of elasticity 55 N
Calculate the elastic potential energy stored in the spring when the extension of the spring is 0.08 metres. Circle your answer. \(0.176 \mathrm {~J} \quad 0.44 \mathrm {~J} \quad 0.88 \mathrm {~J} \quad 1.76 \mathrm {~J}\)
AQA Further Paper 3 Mechanics 2019 June Q2
1 marks Easy -1.8
2 A particle has an angular speed of 72 revolutions per minute.
Find the angular speed in radians per second.
Circle your answer.
[0pt] [1 mark] \(\frac { 6 \pi } { 5 } \quad \frac { 12 \pi } { 5 } \quad 12 \pi \quad 24 \pi\)